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Convex Relaxations for Isometric and Equiareal NRSfM

Agniva Sengupta, Adrien Bartoli

TL;DR

This work proposes convex Relaxations of the isometric model up to quasi-isometry and convex relaxations involving the equiareal deformation model, which preserves local area and has not been used in NRSfM.

Abstract

Extensible objects form a challenging case for NRSfM, owing to the lack of a sufficiently constrained extensible model of the point-cloud. We tackle the challenge by proposing 1) convex relaxations of the isometric model up to quasi-isometry, and 2) convex relaxations involving the equiareal deformation model, which preserves local area and has not been used in NRSfM. The equiareal model is appealing because it is physically plausible and widely applicable. However, it has two main difficulties: first, when used on its own, it is ambiguous, and second, it involves quartic, hence highly nonconvex, constraints. Our approach handles the first difficulty by mixing the equiareal with the isometric model and the second difficulty by new convex relaxations. We validate our methods on multiple real and synthetic data, including well-known benchmarks.

Convex Relaxations for Isometric and Equiareal NRSfM

TL;DR

This work proposes convex Relaxations of the isometric model up to quasi-isometry and convex relaxations involving the equiareal deformation model, which preserves local area and has not been used in NRSfM.

Abstract

Extensible objects form a challenging case for NRSfM, owing to the lack of a sufficiently constrained extensible model of the point-cloud. We tackle the challenge by proposing 1) convex relaxations of the isometric model up to quasi-isometry, and 2) convex relaxations involving the equiareal deformation model, which preserves local area and has not been used in NRSfM. The equiareal model is appealing because it is physically plausible and widely applicable. However, it has two main difficulties: first, when used on its own, it is ambiguous, and second, it involves quartic, hence highly nonconvex, constraints. Our approach handles the first difficulty by mixing the equiareal with the isometric model and the second difficulty by new convex relaxations. We validate our methods on multiple real and synthetic data, including well-known benchmarks.
Paper Structure (13 sections, 2 theorems, 18 equations, 5 figures, 2 tables)

This paper contains 13 sections, 2 theorems, 18 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Methods
  4. Notation and Preliminaries
  5. Problem Statement
  6. Isometric Cost: Relaxation, Convex Reparameterisation and Isometric Methods
  7. Parameterisation with
  8. Parameterisation with
  9. Equiareal Cost: Convex Relaxations and Quasi-equiareality
  10. Parameterisation with
  11. Parameterisation with
  12. Experimental Results
  13. Conclusion

Key Result

Lemma 1

There does not exist a unique solution, even up to scale, for zeroth-order using just equiareal and reprojection constraints.

Figures (5)

  • Figure 1: Observation of the solution spaces for three simulated points in general configuration observed by a perspective camera and parameterised by their depths $\delta_j$, $\delta_q$, and $\delta_r$. The point is in the centre. The solution spaces for inextensibility and equiareality are unbounded subspaces. Using the MDH with inextensibility yields a solution point reasonably close to the . However, using the MDH with equiareality yields a spurious solution point. The proposed quasi-isometric and quasi-equiareal formulations are smaller bounded subspaces enclosing the .
  • Figure 2: We show the following: (a) the variation of w.r.t increasing neighbors in $\mathcal{E}_2$, (b) the variation of w.r.t additive pixel noise to correspondences, (c) the variation of , in , w.r.t increasing $m$ for same $\mathcal{E}_2$ (or ) for all compared methods, (d) the variation of in observed and missing correspondences completed by , (e) the variation of $g\mathrm{E}$ and $a\mathrm{E}$ w.r.t changes in $\lambda_I$ and $\lambda_E$, (f) the variation of in w.r.t increasing $\chi_{\mathrm{E}}$ in , (g) and (h) for each frame from and respectively.
  • Figure 3: Results from rigid dataset.
  • Figure 4: Qualitative results from . Black points are .
  • Figure 5: Non-isometric NRSfM applied to a colonoscopy sequence of 67 images with 72 point tracks. The camera is shown in blue and the reconstructed points are colour-coded according to depth.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Proposition 1
  • proof